From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6938 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: Reference requested Date: Sat, 01 Oct 2011 13:46:33 -0400 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1317581352 10444 80.91.229.12 (2 Oct 2011 18:49:12 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 2 Oct 2011 18:49:12 +0000 (UTC) To: Original-X-From: majordomo@mlist.mta.ca Sun Oct 02 20:49:07 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.30]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RAR62-0007oh-7s for gsmc-categories@m.gmane.org; Sun, 02 Oct 2011 20:49:06 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52074) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RAR48-00034s-Ou; Sun, 02 Oct 2011 15:47:08 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RAR47-00029q-5O for categories-list@mlist.mta.ca; Sun, 02 Oct 2011 15:47:07 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6938 Archived-At: Peter May, in re the Subject: categories: Re: Reference requested, wrote > ... I prefer `chaotic' to `indiscrete' not just because > of the `coarse' implications of the latter, but because > indiscrete spaces are boring, `null or banal', whereas > chaotic categories have genuinely significant applications. ... Be that as it may, I sought Search-engine advice regarding the use of the= = 'chaotic topological space' lingo, and came up with the following 'hits',= of which only the first reflects, in an afterthought, Peter's usage, while the others all envision something rather quite different: 1) From http://en.wikipedia.org/wiki/Grothendieck_topology : = The discrete and indiscrete topologies Let C be any category. To define the discrete topology, we declare all si= eves to be covering sieves. If C has all fibered products, this is equivalent = to declaring all families to be covering families. To define the indiscrete topology, we declare only the sieves of the form Hom(=E2=88=92, X) to be = covering sieves. The indiscrete topology is also known as the biggest or chaotic topology, and it is generated by the pretopology which has only isomorphi= sms for covering families. A sheaf on the indiscrete site is the same thing a= s a presheaf. Other uses of 'chaotic', having nothing to do with indiscreteness, predominate: 2) From http://www.math.uh.edu/~hjm/pdf26%284%29/03chara.pdf , reflecting the content of = ON GENERALIZED RIGIDITY by JANUSZ J. CHARATONIK from Houston Journal of Mathematics (© 2000 University of Houston) Volume 26, No. 4, 2000 : A nondegenerate topological space X is said to be: (a) chaotic if for any two distinct points p and q of X there exists an o= pen neighbourhood U of p and an open neighbourhood V of q such that no open subset of U is homeomorphic to any open subset of V ; ... [snip] ... 3) CHAOTIC GROUP ACTIONS www.math.zju.edu.cn/amjcu/B/200301/030108.pdf =2E.. no chaotic group actions on any topological space with free arc. ..= =2E =2E.. topological space which admits a chaotic group action but admits ..= =2E 4) CHAOTIC POLYNOMIALS IN SPACES OF CONTINUOUS AND ... personales.upv.es/almimon/Preprint%20Aron-Miralles.pdf =2E.. show that there exist chaotic homogeneous polynomials of degree m =E2= =89=A5 2. =2E.. So I'd imagine 'chaotic', for 'indiscrete', is best dropped, and either 'indiscrete', 'codiscrete', or 'trivial' be used instead. NB: while it's true that the trivial (indiscrete) topology on a set X = is initial, in the sense that, as a collection of subsets of X, it's the = smallest that's a topology on X, the indiscrete topological space on X is terminal among all topological spaces on X and mappings that restrict to the identity on X; the trivial (indiscrete) pre-order on X is likewise= terminal, in the sense that, as a subset of X x X, it's the largest. A connected pre-ordered groupoid (i.e., indiscrete category), being equivalent to the terminal category 1, has the property that, for each category X, it admits exactly one isomorphism class of functor from X, but while that may make it 2-terminal or [( co | op ) lax-] terminal, = I'd still probably prefer to avoid such ... umm ... terminalogy :-) . Cheers, -- Fred PS: I re-emphasize: of all the hits I found, only one amongst the first two dozen -- the first cited above -- spoke of the trivial topology as = the chaotic topology; ALL the others used 'chaotic' in some other way, = DESPITE the search having been explicitly for [ chaotic topological space= ]. = And there were "about 175,000 results" all told :-) . -- F. PPS: Typos? Perhaps; please forgive, I couldn't spiel-chuck. -- F. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]